XVA Analysis From the Balance SheetStéphane Crépey
https://math.maths.univ-evry.fr/crepey
LaMME, Univ Evry, CNRS, Université Paris-Saclay
Cours Bachelier, IHP, janvier–février 20181 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

Since the 2008 crisis, investment banks charge to their clients, in theform of rebates with respect to the counterparty-risk-free value offinancial derivatives, various add-ons meant to account for
counterparty risk and its capital and funding implications.
These add-ons are dubbed XVAs, where VA stands for valuation
adjustment and X is a catch-all letter to be replaced by C for credit,
D for debt, F for funding, M for margin, K for capital (!), and so on.

3 / 244

Pricing XVA add-ons at trade levelfunds transfer price (FTP)

But also accounting XVA entries at the aggregate portfolio level

In June 2011 the Basel Committee reported that
During the financial crisis, roughly two-thirds of losses attributed
counterparty credit risk were due to CVA losses and only about on
third were due to actual defaults
In January 2014 JP Morgan has recorded a $1.5 billion FVA loss

Individual FTP of a trade actually computed as portfolio incremental
XVAs of the trade

4 / 244

XVAs deeply affect the derivative pricing task by making it global,
nonlinear, and entity-dependent
But, before coming to the technical implications, the fundamental
points are:
1. To understand what deserves to be priced and what does not⊇ Double counting issues

2. To establish not only the pricing, but also the corresponding
collateralization, accounting, and dividend policy of the bank5 / 244

Hull and White (2012) Risk Magazine notes, although one need notagree with their conclusions that rely on perfect market assumptions,
had the merit to shift the debate to the right grounds:
Properly accounting for the misalignement of interest between theshareholders and creditors and the bankAnd, the way we put it, for the coexistence and interaction within thebank of XVA versus “clean” desks.

6 / 244References

Own:
C. Albanese and S. Crépey. XVA Analysis From the Balance Sheet.
Working paper 2017.
C. Albanese, S. Caenazzo and S. Crépey. Credit, Funding, Margin, andCapital Valuation Adjustments for Bilateral Portfolios. Probability,
Uncertainty and Quantitative Risk 2017.

7 / 244Others:
On the funding issue,
Burgard and Kjaer (BK) FVA papers,
Andersen, L., D. Duffie, and Y. Song (2016). Funding value
adjustments. ssrn.2746010;

On the capital issue,
The Solvency II actuarial literature,
Green and K enyon (GK) K VA papers.

8 / 244The XVA benchmark model
BK and GK papers(dozens of thousands of downloads on ssrn and arxiv!!) illustrate
their points using a Black–Scholes model S for an underlying market
risk factor, in conjunction with independent Poisson counterparties
and bank defaults
Warning: using a Black–Scholes (replication) framework as an XVA
toy model is convenient.
But, as FVA and KVA are mostly about market incompleteness in our
opinion, there are some pitfalls to it.

9 / 244The XVA benchmark model (cont’d)
BK advocate a replication XVA approach and blame risk-neutral
approaches outside the realm of replication (see the first paragraph intheir 2013 paper).
But BK papers themselves end-up doing what they callsemi-replication, which is nothing but a form of risk-neutral pricing
without (exact) replication.
Addressing the KVA by replication, as done under the GK approach,is a bit of a contradiction in terms.
More detailed comparison and comments in later sections of theseslides.

10 / 244

Synopsis

Assumption 1
For accepting a new deal, shareholders need be at least indifferent given
the cash flows before τ only.
From this point of view, a key distinction is between
the cash flows received by the bank prior its default time τ
the cash flows received by the bank during the default resolution period
starting at τ .

The first stream of cash flows affects the bank shareholders, whereas
the second stream of cash flows only affects creditors.

11 / 244

Assumption 2
A bank cannot replicate jump-to-default related cash-flows.
Own default related cash-flows, in particular
See Castagna and Fede (2013, Section 10.7)

12 / 244

1. Based on this market incompleteness tenet, we show that the all
inclusive XVA add-on (funds transfer price FTP) that aligns derivative
prices to the interest of bank shareholders is
FTP = CVA + FVA + KVA,

a. to be contrasted with the “fair valuation” (CVA − DVA) of
counterparty risk,b. where each XVA term is nonnegative (and solves a nonstandard
backward SDE stopped before the bank default time τ ).

13 / 244

2. Meant incrementally at every new deal, the above XVA add-on will beinterpreted dynamically as the cost of the possibility for the bank to
go into run-off,i.e. lock its portfolio and let it amortize in the future,

while staying in line with shareholder interest, from any point in time
onward if wished.

14 / 244

→ Basel III Pillar 2 FTP as a “soft landing” or “anti-Ponzi” corrective
pricing scheme accounting for counterparty risk incompleteness:

15 / 244

Contra-assets and Contra-liabilities
CA: Contra-assets, entail the valuation of all cash-flows related to thecredit risk of either the counterparties or the bank and occurring
before the default of the bank itself, i.e. having an impact onshareholder value.
CVA, FVA, ...

CL: Contra-liabilities, entail the valuation of all the cash-flows received by
the bank during the resolution process starting at its default time ,i.e. only having an impact on bank creditors, by modifying therecovery rate of the bank, but not on shareholders.
DVA, FDA, ...

16 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

In this section we present the main ideas of our XVA approach in anelementary static one-year setup, with r set equal to 0.
Assume that at time 0 a bank, with equity E = w0 corresponding to
its initial wealth, enters a derivative position (or portfolio) with aclient.
Let P = EP denote the mark-to-market of the deal ignoring
counterparty risk and assuming risk-free funding.

18 / 244

We assume that the bank and its client are both default prone withzero recovery.
We denote by J and J1 the survival indicators of the bank and its
client at time 1
Both being assumed alive at time 0With default probability of the bank Q(J = 0) = γ
And no joint default for simplicity, i.e Q(J = J1 = 0) = 0.

19 / 244

XVA Cost of Capital Pricing Approach

In order to focus on counterparty risk and XVAs, we assume that themarket risk of the bank is perfectly hedged by means of perfectly
collateralized back-to-back trades
The back-to-back hedged derivative portfolio reduces to its
counterparty risk related cash flows

At the bottom of this work lies the fact that a bank cannot replicate
jump-to-default exposures
Cost of capital pricing approach in incomplete counterparty risk
markets

20 / 244

Standing risk-neutral valuation measure Q
Derivative entry prices in our sense include, on top of the valuation of
the corresponding cash-flows, a KVA risk premium
Risk margin (RM) in a Solvency II terminology
Computed assuming P = Q, as little of relevance can be said about thehistorical probability measure for XVA computations entailingprojections over decades
The discrepancy between P and Q is left to model risk

Cost of capital pricing approach applied to the counterparty risk
embedded into the derivative portfolio of a bank

21 / 244Cash FlowsThe counterparty risk related cash flows affecting the bank before its
default are its counterparty default losses and funding expenditures,
respectively denoted by C ◦ and F ◦ .
The bank wants to charge to its client an add-on, or obtain from its
client a rebate,depending on the bank being “seller or buyer”,

denoted by CA, accounting for its expected counterparty default
losses and funding expenditures.

22 / 244

Accounting for the to-be-determined add-on CA, in order to enter the
position, the bank needs to borrow (P − CA)+ unsecured or invest(P − CA)− risk-free, depending on the sign of (P − CA), in order topay (P − CA) to its client.

23 / 244

At time 1:
If alive (i.e. J = 1), then the bank closes the position while receiving P
if its client is alive (i.e. J1 = 1) or pays P − if its client is in default(i.e. J1 = 0).
Note J1 P − (1 − J1 )P − = P − (1 − J1 )P + . Hence the counterparty
default loss of the bank appears as the random variableC ◦ = (1 − J1 )P + .

(1)

In addition, the bank reimburses its funding debt (P − CA)+ or
receives back the amount (P − CA)− it had lent at time 0.
If in default (i.e. J = 0), then the bank receives back P + on thederivative as well as the amount (P − CA)− it had lent at time 0.

24 / 244

We assume that unsecured borrowing is fairly priced as γ × theamount borrowed by the bank, so that the funding expenditures ofthe bank amount toF ◦ = γ(P − CA)+ ,deterministically in this one-period setup.
We assume further that a fully collateralized back-to-back market
hedge is set up by the bank in the form of a deal with a third party,
with no entrance cost and a payoff to the bank −(P − P) at time 1,
irrespective of the default status of the bank and the third party attime 1.

25 / 244

Collecting cash flows, the wealth of the bank at time 1 is
w1 = E − F ◦ + (1 − J) P + + (P − CA)−+J J1 P − (1 − J1 )P − − (P − CA)+ + (P − CA)− − (P − P)
= E − (C ◦ + F ◦ − CA) + (1 − J)(P − + (P − CA)+ ),(2)

as easily checked for each of the three possible values of the pair (J, J1 )26 / 244

The result of the bank over the year is
w1 − w0 = w1 − E = −(C ◦ + F ◦ − CA)+ (1 − J)P − + (1 − J)(P − CA)+ .
{z}| {z } |C•

F•

However, the cash flow (C • + F • ) is only received by the bank if it isin default at time 1, so that it only benefits bank creditors.
Hence, the profit-and-loss of bank shareholders reduces to−(C ◦ + F ◦ − CA), i.e. the trading loss-and-profit of the bank, which
we denote by L, appears as
L = C ◦ + F ◦ − CA.

(3)

27 / 244

Remark 1
The derivation (2) allows for negative equity, which is interpreted asrecapitalization.
In a variant of the model excluding recapitalization, where the defaultof the bank would be modeled in a structural fashion as E − L < 0and negative equity is excluded, we would get instead of (2)w1= (E − L)+ + 1{E<L} (P − + (P − CA)+ ).

(4)

In our approach we consider a model with recapitalization for reasons
explained later.

28 / 244

CVA and FVA

In order to balance the trading loss and profit L, the bank charges to
its client the add-on◦
CA = |{z}EC ◦ + EF
|{z},
CVA

(5)

FVA

which accounts for the expected counterparty default losses andfunding expenditures of the bank.

29 / 244

Note that, since
FVA = EF ◦ = F ◦ = γ(P − CA)+
(all deterministically in a one-period setup), (5) is in fact an equation
for CA.

30 / 244

Equivalently, we have the following semi-linear equation for
FVA = CA − CVA :FVA = γ(P − CVA − FVA)+ ,which has the unique solution
FVA =γ(P − CVA)+ .
1+γ

31 / 244

Substituting this and (1) into (5), we obtainγ
CA = E[(1 − J1 )P + ] +(P − CVA)+ .
|
{z
} 1+γ
{z
}
|
CVA

(6)

FVA

32 / 244

KVA and Funds Transfer Price

Note that the realized recovery isC • + F • = (1 − J)(P − + (P − CA)+ )because of the trade that occurred, but this was not anticipated andnot reflected in the price of borrowing when the bank issued its
funding debt.
As the funding debt was fairly valued ignoring this, the value
FDA = E[(1 − J)(P − CA)+ ] of the default funding cash flow F •equals the cost FVA = γ(P − CA)+ of funding the position.

33 / 244

But the FVA and the FDA do not impact the same economic agent,
namely the FVA hits bank shareholders whereas the FDA benefits
creditors.
Hence, the net effect of funding is not nil to shareholders, but reduces
to an FVA cost.

34 / 244

In view of (3) and (5), observe that charging to the client a CA
add-on corresponding to expected counterparty default losses andfunding expenditures is equivalent to setting the add-on CA suchthat, in expectation, the trading loss-and-profit of bank shareholders
is zero (EL = 0), as it would also be the case without the deal.
However, without the deal, the loss-and-profit of bank shareholders
would be zero not only in expectation, but deterministically.

35 / 244

Hence, to compensate shareholders for the risk on their equity
triggered by the deal, under our cost of capital approach, the bankcharges to its client an additional amount (risk margin)
KVA = hE,(7)

where h is some hurdle rate, e.g. 10%.

36 / 244

Moreover, since E can be interpreted as capital at risk earmarked toabsorb the losses (C ◦ + F ◦ ) of the bank above CA, it is natural to size
E by some risk measure of the bank shareholders loss-and-profit L.
The all-inclusive XVA add-on to the entry price for the deal, which wecall funds transfer price (FTP), is
FTP =

CA
+
KVA
.
|{z}
| {z }
Expected costs Risk premium(8)37 / 244

Monetizing the Contra-Liabilities?

Let us now assume, for the sake of the argument, that the bankwould be able to hedge its own jump-to-default risk through a further
deal, whereby the bank would deliver a payment (C • + F • ) at time 1
in exchange of an upfront fee fairly valued as
CL = |{z}EC • +
DVA

•
EF
|{z}

FDA=γ(P−CA)+ =FVA.

(9)

DVA and FDA stand for debt valuation adjustment and funding debt
adjustment.

38 / 244

Let CR denote the modified charge to be passed to the client when
the hedge is assumed.
Accouting for the hedging gain Hcl = CL − (C • + F • ), the wealth of
the bank at time 1 is now given by (cf. (2))
w1

= (E − (C ◦ + F ◦ − CR)) + C • + F • + Hcl= E − (C ◦ + F ◦ − CR) + CL.

(10)

39 / 244

By comparison with (2), the CL originating cash flow (C • + F • ) is
hedged out and monetized as an amount CL received by the bank attime 0.
The trading loss-and-profit of bank shareholders now appears as
L = w0 − w1 = E − w1 = C ◦ + F ◦ − CR − CL.

40 / 244

The amount CR making L centered (and actually the same as before)is
CR = E(C ◦ + F ◦ ) − CL
= (CVA + FVA) − (DVA + FDA) = CVA − DVA,

(11)

because FVA=FDA (cf. (9)).
Hence, if the bank was able to hedge its own jump-to-default risk, inorder to satisfy its shareholders in expectation, it would be enough forthe bank to charge to its client an add-on CR = CVA − DVA.

41 / 244

The amount CR = CVA − DVA can be interpreted as the fairvaluation of counterparty risk when market completeness and no
trading restrictions are assumed (cf. Duffie and Huang (1996)).
However, under our approach, in the present setup, the bank would
still charge to its client a KVA add-on hE as risk compensation for
the non flat loss-and-profit L triggered by the deal (with E sized bysome risk measure of L).

42 / 244

Wealth Transfer Interpretation

We see from (5) and (9) that CA can be viewed as the sum between
CL and the fair valuation CR = CVA − DVA of counterparty risk.
CL can be interpreted as an add-on that the bank needs to sourcefrom the client, on top of the fair valuation of counterparty risk, inorder to compensate the loss of value to shareholders due to theinability of the bank to hedge its own jump-to-default risk.

43 / 244

Due to this market incompleteness (or trading restriction), the deal
triggers a wealth transfer from bank shareholders to creditors equal toCL, which then needs be sourced by the bank from its client in orderto put shareholders back at value equilibrium in expected terms.

44 / 244

Funds Transfer Price Decomposition

In conclusion, in a one-period setup, the FTP can be represented as
FTP =
=

KVA
CVA
+ FVA} +| {z }{z|
Expected costs CA Risk premium+ FDA} +KVA,
CVA− DVA} + DVA| {z }{z|{z|
Fair valuation CR Wealth transfer CL Risk premium(12)

where the CVA an the FVA are detailed in (6) and where the randomvariable L used to size the equity E in the KVA formula (7) is thebank shareholders loss-and-profit L as per (3).

45 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

Needof a multi-period model, which involves rebalancing between various
banking accounts, for dealing with dynamic portfolios,
to cast the so-called contra-assets and contra-liabilities of the bank in abalance sheet perspective, in order to obtain not only the (entry) prices
view on XVAs, but also the XVA accounting perspective→ We introduce a capital structure model of a bank that shows thedifferent bank accounts involved.

47 / 244

Balance Sheet of a BankASSETS AND CONTRA-LIABILITIESContra-liabilities
CL=DVA+FDA+CVA-CL+ FVA-CL

Shareholder capital at risk (SCR)
Accounting equity

Economic capital (EC)Core equity tier I capital (CET1)

Risk Margin
RM=KVAKVA desk(management)

CONTRA-ASSETS AND LIABILITIESFVACA desks

yr1
Clean desks

FVA desk

Contra-assets (CA)Reserve capital (RC)yr39 yr40

Mark-to-market of thefinancial derivatives (MtM)

yr1(Treasury)
CVA desk

CVA
yr39

yr40

Collateral received by theclean desks (clean margin CM)

48 / 244

A Bank With Three Floors

We consider a central “CA desk” of the bank, in charge of absorbing
counterparty default losses and funding expenditures.
But not of the KVA, which is treated separately.

49 / 244

The “CA = CVA + FVA desk(s)” of the bank sells the corresponding“contra-assets” to the clients of the bankPuts the ensuing “reserve capital” in an RC account, which is thenused for absorbing the counterparty default losses and risky funding
expenditures of the bank

50 / 244

After the contracts have thus been cleaned of their counterparty riskand (other than risk-free) funding implications by the CA desk, theother trading desks of the bank, which we call clean desks (or“bottom floor”) of the bank, are left with the management of themarket risk of the contracts in their respective business lines, ignoring
counterparty risk.

51 / 244

The top (third) floor is the management in charge of the KVA
payments, i.e. of the dividend distribution policy of the bankPuts the “risk margin” sourced from the clients in an RM account,which is then gradually released to shareholders as a remuneration for
their capital at risk

52 / 244

CA desk cash flows graph: contractually promised P, risky fundingF, and hedging H cash flows
Clients

Financial markets
P

HC

CA deskF
P

Clean desks

Bank creditors

53 / 244

In fact, we deal with two portfolios, the client portfolio between theclients of the bank and the CA desk and the cleaned portfolio
between the CA desk and the clean desks.
The corresponding (cumulative streams of) contractually promised
cash flows are the same, denoted by P. But, as intuitively clear anddetailed in the sequel, counterparty risk only really impacts the client
portfolio.

54 / 244

Pricing, Accouting and Dividend Policies of the Bank

Another view on “mark-to-model”:
The RM account is continuously reset by the management of thebank to its (to be determined) theoretical target KVA level.
(−dKVAt ) amounts continuously flow from the RM account to theshareholder dividend stream

The RC account is continuously reset by the CA desk to its (to bedetailed) theoretical target CA = CVA + FVA level(−dCAt ) amounts continuously flow from the RC account to theshareholder dividend stream

55 / 244

The CM account is continuously reset by the clean desks of the bankto the MtM value of the portfolio (ignoring counterparty risk, which
they do not even “see” in their setup)(−dMtMt ) amounts continuously flow from the CM account to theclean desks trading loss and profit process→

Balance conditions
CM = MtM , RC = CA = CVA + FVA , RM = KVA(13)

56 / 244

Assumption 3
The activity of each (clean or CA) trading desk, hence of the bank as awhole, is self-financing.
At this stage this is only a broad axiom, already implicitly postulated
in the one-period setup (cf. (2)), by which we mean that the wealthof a trading desk at a later time results from the development of its
wealth at an earlier time as the sole consequence of its pricing,hedging, funding, and collateralization policy, and of the evolution of
the underlying risk factors, without any further creation or
annihilation of cash flows.

57 / 244

Bank Default Model

Instead of viewing losses as money flowing away from the balance
sheet, we view them as money flowing into it as refill ,i.e. replenishment of the different bank accounts at their theoreticaltarget level,
until the point of default where the payers cease willing to do so .

58 / 244

When this happens is modeled as a totally unpredictable time τcalibrated to the bank CDS spread , which we view as the mostreliable and informative credit data regarding anticipations of markets
participants about future recapitalization, government intervention
and other bank failure resolution policies.

59 / 244

Comparison with the Merton model
In a Merton mindset, the default of the bank in our setup would bemodeled as the first time when the core equity (CET1) of the bankbecomes negative, where CET1 depletions correspond to the banktrading losses L
In the case of a bank, given recapitalisation and managerial resolution
schemes, CET1 is constantly “refilled” by the shareholders and it ismore realistic to model the default as a totally unpredictable (liquidity
or operational) event at some exogenous time τ with intensity γ
calibrated to the bank CDS spread.
cf. Duffie (2010)’s analysis of major bank defaults during the crisis

60 / 244

Comparison with the Merton model (Cont’d)
The purpose of our capital structure model of the bank is not tomodel the default of the bank as the point of negative equity, which
would be unrealistic (and in fact never occurs in our setup)
Instead, our aim is to put in a balance sheet perspective thecontra-assets and contra-liabilities of the bank, items which are notpresent in the Merton model.

61 / 244

Invariance Valuation Setup

The clean desks of the bank, who are immunized against counterparty
risk through the action of the CA desk, typically ignore the default of
the bank in their modeling.
They price and hedge in order to be non arbitrable ignoring thedefault of the bank, using some reference filtration F such that τ is
not an F stopping time.

62 / 244

But the bank is defaultable, hence the full model information used by
the CA desk, as well as by the management of the bank in charge ofthe KVA payments, is a larger filtration G such that τ is a G stoppingtime.

Assumption 4
Any G stopping time η admits an F stopping time η 0 such thatη ∧ τ = η 0 ∧ τ ; any G semimartingale Y admits a unique F semimartingale
Y 0 , called the reduction of Y , that coincides with Y before τ .

63 / 244

We denote by J = 1[0,τ ) the survival indicator process of the bank.
For any left-limited process Y , we denote by
Y ◦ = Y τ − = JY + (1 − J)Yτ −the process Y stopped before time τ and we write Y • = Y ◦ − Y .
Accounting for the bank default time τ , the time horizon of themodel is τ̄ = τ ∧ T , where T is the final maturity of all claims in theportfolio, (first) assumed held on a run-off basis.

64 / 244

Definition 1
By trading loss Lcl of the clean desks, we mean the negative of theirwealth process W cl , as the latter results from their trading by anapplication of the self-financing assumption defined with respect tothe reference filtration F.
By trading loss Lca of the CA desk, we mean the negative of its wealthprocess W ca , as the latter results from its trading by an application of
the self-financing assumption with respect to the filtration G, stopped
before τ for alignment with shareholder interest. That is,
Lca = −(W ca )◦ .

(14)

65 / 244

Definition 1 (Cont’d)
By trading loss L of the bank as a whole, we mean
L = −(W cl + W ca )◦ = (Lcl )◦ + Lca .

(15)

66 / 244

Consistency of valuation across the perspectives of the different desksof the bank is granted by the following:

Assumption 5
Clean desks and CA desks generate trading losses that are martingaleswith respect to their respective pricing bases (F, P) and (G, Q),
where the latter are such that (F, P) local martingales stopped before
τ are (G, Q) local martingales,
whereas the reductions of (G, Q) local martingales stopped before τare (F, P) local martingales.

67 / 244

i.e. τ is an invariance time as per Crépey and Song (2017a).
Standard (but by no means limitative) situation: immersion setup
where (F, Q) local martingales are (G, Q) local martingales without
jump at τ , in which case τ is an invariance time with P = Q.

68 / 244

As an immediate consequence of Assumption 5:

Corollary 1
The trading loss Lcl of the clean desks is an (F, P) local martingale.
The trading losses Lca and L = Lca + (Lcl )◦ of the CA desk and of thebank as a whole are (G, Q) local martingales without jump at time τ , their
reductions are (F, P) local martingales.

69 / 244

Assumption 6
The bank cannot hedge its own jump-to-default exposure, hence H = H◦ .

70 / 244

Given Y representing a process of cumulative cash flows or trading or
hedging losses, respectively an XVA process, we denote by Ye thecorresponding process of cumulative OIS discounted cash flows or tradingor hedging losses, respectively the corresponding OIS discounted XVA
process.

Example 1Le =R·

0 βt dLt , CVA = βCVA.

]

71 / 244

Corollary 2
In the case of a cumulative cash flow or loss process Y , the process Y is alocal martingale if and only if Ye is a localmartingale.

72 / 244

Loss and Dividend Processes

Lemma 1
The trading loss processes Lca of the CA desk and L of the bank as awhole, both risk neutral local martingales by application of Corollaries 1
and 2, are given byg◦ + Ce◦ + Fe◦ + H
eLeca = CAg◦ + Ce◦ + Fe◦ + H
e + (Lecl )◦ ,Le = CA(16)

73 / 244

Lemma 2
Shareholder cumulative discounted dividends are given by◦] .
−Le − KVA

(17)

We emphasize that, in our model, negative dividends are possible. They
are interpreted as recapitalisation, i.e. equity dilution.

74 / 244

All our XVA processes are sought for in a suitable Hilbert space S2 ofsquare integrable G adapted processes containing the null process,defined until time τ̄
(G, Q) valuation is never needed beyond that point.

We denote by S2◦ the corresponding subspace of processes Y without
jump at τ and such that YT = 0 on {T < τ }.

75 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

Clean valuation and mark-to-market of the portfolio

Definition 2
Given an F adapted cumulative cash flow stream D, the OIS
discounted (F, P) value process of D is the (F, P) conditional
expectation process of the future OIS discounted cash flows in D.
By clean valuation of a contract (or portfolio) with F adapted
contractually promised cash flow stream D, we mean the (F, P) value
process of D.

77 / 244

Proposition 1
Clean valuation is additive over contracts, i.e. the mark-to-market of
a portfolio of contracts is the sum of the mark-to-markets of theindividual contracts.
Clean valuation is also intrinsic to the contracts themselves. In
particular, it is independent of the involved parties and of theircollateralization, funding and hedging policies.
Proof. The promised cash flows of a portfolio are the sum of thepromised cash flows of the contracts. Hence the result follows by linearityof the cash flow (F, P) valuation rule of Definition 2.

78 / 244

Consistently with the (F, P) local martingale condition on the trading loss
process of the clean desks:

Definition 3
By mark-to-market MtM of the (cleaned or client) portfolio, we mean theclean valuation of the contracts not yet liquidated in the portfolio (i.e.
ignoring counteparty risk for the future).

79 / 244

Valuation of the Funding and Hedging Assets

Definition 4
Given a G adapted cumulative cash flow stream D, the OIS discounted(G, Q) value process of D is the (G, Q) conditional expectation process ofthe future OIS discounted cash flows in D.

80 / 244

Assumption 7
The funding costs of the CA desk are of the form(−OIS accrual of the RC account) + F,

(18)

for some (G, Q) martingale F starting from 0 with nondecreasing F ◦ andF • components.

81 / 244

The first term in (18) is the funding benefit to which funding would
boil down if risk-free funding was available to the bank.
The rationale underlying Assumption 7 is that funding is implemented
in practice as the stochastic integral of predictable hedging ratios
against traded funding assets.

82 / 244

Under the cash flow (G, Q) valuation rule of Definition 4, the value
process of each of these assets is a martingale modulo risk-free
accrual.
Therefore the funding costs of the bank accumulate into a (G, Q)
martingale F, coming on top of a risk-free accrual (actual benefit,
i.e. negative cost) of the RC cash account of the CA desk.

83 / 244

The assumption that F ◦ is nondecreasing rules out models where thebank can invest (not only borrow) at his unsecured borrowing spreadover OIS, because, as a consequence on F • through the martingale
condition on the process F as a whole, this would imply that thebank can hedge its own jump-to-default exposure.
That is, we assume only windfall at bank own default, no shortfall
cf. the detailed discussion of BK papers later in these slides.

84 / 244

In view of Assumption 6, this also insures some kind of orthogonality
between the risky funding and hedging loss martingales F and H, sothat F and H are nonsubstituable to each other (and the bankcannot manipulate by using one for the other).

85 / 244

Example 2Let
dBt = rt Bt dt

dDt = (rt + λt )Dt dt + (1 − R)Dt− dJt = rt Dt dt + Dt− λt dt + (1 − R)drepresent the risk-free OIS deposit asset and a risky bond issued by
the bank for its investing and unsecured borrowing purposes.
λ represents the unsecured funding spread of the bank and R is thecorresponding recovery coefficient, taken as an exogenous constant.

86 / 244

Example 2 (Cont’d)
The risk-neutral martingale condition that applies to (βD) under ourstanding valuation framework implies that λ = (1 − R)γ, hence
λt dt + (1 − R) dJt = (1 − R)dµt ,where dµt = γdt + dJt is the (G, Q) compensated jump-to-default
martingale of the bank87 / 244

Example 2 (Cont’d)
We assume all re-hypothecable collateral and we denote by Q theamount of collateral posted by the CA desk to the clean desks net ofthe amount received by the CA desk from the clients.

88 / 244

Example 2 (Cont’d)
The funding policy of the CA desk is represented by a splitting of theamount CAt on the RC account of the bank as
CAt =

Qt
|{z}
Collateral remunerated OIS+(CA − Q )+| t {z t }Cash in excess invested at the risk-free rate−
(CAt − Qt )−|{z}
Cash needed unsecurely funded=Qt + (CAt − Qt )+−(CAt − Qt )−
|
{z
}
|
{z
}
Invested at the risk-free rate as νt Bt Unsecurely funded as ηt D

89 / 244

Example 2 (Cont’d)
A standard continuous-time self-financing equation expressing theconservation of cash flows at the level of the bank as a whole yieldsd (νt Bt − ηt Dt ) = νt dBt − ηt− dDt= νt rt Bt dt − ηt (rt + λt )Dt dt − (1 − R)ητ − Dτ − dJt(20)= rt CAt dt − (1 − R)ηt− Dt− dµt , 0 ≤ t ≤ τ̄A left-limit in time is required in η because D jumps at time τ, so thatthe process η, which is defined implicitly through (CA − Q)− /D in(19), is not predictable.

90 / 244

Example 2 (Cont’d)Equivalently viewed in terms of costs, i.e. flipping signs in the above,
we obtain− d (νt Bt − ηt Dt ) = −rt CAt dt + dFt ,where dFt = (1 − R)(Qt− − CAt− )+ dµt , which is in line with
Assumption 7

91 / 244

Regarding now hedging losses:

Assumption 8
The hedging loss H = H◦ of the CA desk, including the cost of setting thehedge, is a (G, Q) local martingale starting from 0.

92 / 244

The rationale here is that hedging gains or losses arise in practice as
the stochastic integral of predictable hedging ratios against wealthprocesses of traded hedging assets.
Note that we are considering wealth processes inclusive of theassociated funding costs here, which corresponds to the mostcommon situation of hedges that are either swapped or tradedthrough a repo market, without upfront payment.

93 / 244

Under the cash flow (G, Q) valuation rule of Definition 4, eachhedging asset is valued as risk-free discounted expectation of its
future cash flows.
Hence the wealth processes related to long positions in any of thehedging assets are (G, Q) local martingales, as are stochastic integrals
against them.

94 / 244

Example 3
Assuming the CA hedge implemented through a repo market on aBlack-Scholes stock S with volatility σ, then, supposing no dividends andno repo basis on S:dHt = −ζt dS − rSt dt = −ζt σSt dWt ,(21)
where W is the (G, Q) Brownian motion driving S and ζ is the hedging
ratio used in S.
The instantaneous cost of funding the hedge is (ζt rSt dt), which isincluded in (21).

95 / 244

Remark 2
The valuation impact of a theoretical (but impractical) hedge by the bankof its contra-liabilities will be considered separately in Proposition 3.
As immediate consequences of Assumptions 7 and 8 :Corollary 3
The processes F and H = H◦ are (G, Q) martingales with zero (G, Q)
value.

96 / 244

Contra-assets

Proposition 2
CA, CVA, and FVA are the solutions to the following fixed-point
problems, assumed well-posed in S2◦ : For t ≤ τ̄ ,gt = Et Ceτ̄◦ + Feτ̄◦ − Cet◦ − Fet◦ + 1{τ <T } CA
g◦τ
CA




f ◦τ ,= Et Ceτ̄◦ + Feτ̄◦ − Cet◦ − Fet◦ + 1{τ <T } RC

(22)

by (13), and◦] t = Et Ceτ̄◦ − Cet◦ + 1{τ <T } CVA
]τ ,
CVA
◦
] t = Et Feτ̄◦ − Fet◦ + 1{τ <T } FVA
]τ .
FVA

(23)
(24)

97 / 244

Proof. By application of the (G, Q) martingale condition on Leca as per(16), combined with Corollary 3.

98 / 244

Remark 3
The industry terminology tends to distinguish an FVA, in thetechnical sense of the cost of funding cash collateral for variation
margin, from an MVA, defined as the cost of funding segregatedcollateral posted as initial margin (see Albanese et al. (2017)).
The academic literature, as in these slides, tends to merge the two inan overall FVA meant in the broad sense of the cost of funding thederivative trading strategy of the bank.

99 / 244

Contra-liabilities

Definition 2 (Cont’d)
By contra-liabilities value process CL, we mean
CL = DVA + FDA + CVACL + FVACL ,(25)

where:The DVA (debt valuation adjustment) is the (G, Q) value of C • ,The FDA (funding debt adjustment) is the (G, Q) value of F • , andCVACL and FVACL are the (G, Q) values of terminal cash flows1{τ <T } CVA◦τ and 1{τ <T } FVA◦τ at time τ̄ .

100 / 244

Financial interpretation
The DVA is the value that the bank clients lose due to the possible
default of the bank in the future.
The FDA is the value of the amount of its funding debt that the bankfails to reimburse if it defaults.
CVACL and FVACL are contra-liability components of the CVA andthe FVA, valuing the residual amounts 1{τ <T } CVA◦τ and1{τ <T } FVA◦τ , summing up to 1{τ <T } CA◦τ = 1{τ <T } RC◦τ (cf. (13)),which are transferred from the RC account to bank creditors at timeτ.

101 / 244

Wealth Transfer Analysis

Let us now assume, for the sake of the argument, that the bankwould be able to hedge its own jump-to-default risk by selling on thefinancial markets a contract paying CLτ at time τ (e.g. throughrepurchasing of its own bond as contemplated in Burgard and Kjaer(2011a, 2011c)).
Let CR denote the modified charge to be passed to the client formaking the trading loss and profit of the bank a (G, Q) martingale,assuming this hedge in place.

102 / 244

Proposition 3
We have
CR = CA − CL,(26)which is also the (G, Q) value of C (or of (C + F + H)).

Corollary 4
CL is interpreted as the wealth transfer triggered by the deals fromshareholders to creditors, due to the inability of the bank to hedge its own
jump-to-default exposure.

103 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

Not only a bank cannot hedge its own jump-to-default: It cannot
replicate its counterparty default losses either.
An XVA add-on defined by CA = CVA + FVA ensures that thetrading loss L of the bank is zero in expectation.
But the impossibility of replicating counterparty default losses implies
that the trading of the bank generates a non-vanishing loss-and-profit
process L.
Then the regulator comes and requires that capital be set at risk by
the shareholders, which therefore require a risk premium.

105 / 244

Valuation is risk-neutral with respect to the stochastic bases (F, P) or(G, Q).
Economic capital and KVA assess risk and its cost, which refer to thehistorical probability measure.

106 / 244

In our setup, the duality of perspective of the clean vs. CA desks, on
pricing as reflected by Assumption 5, also applies to risk
measurement.
Capital calculations are always made “on a going concern”,
i.e. assuming that the bank is alive, and therefore with respect to thereference filtration F.
Instead, cost of capital calculations are made by the management of
the bank using the filtration G including the default of the bank.

107 / 244

However, in the context of XVA computations entailing projections
over decades, the main source of information is market prices of liquid
instruments, which allow the bank to calibrate the pricing measure,and there is little of relevance that can be said about the historicalprobability measure.

108 / 244

Hence, in our model:

Assumption 9
The estimates of the historical probability measure respectively used ineconomic capital and cost of capital computations coincide with thepricing measures P and Q.
Any discrepancy between the historical and risk-neutral measures isleft to model risk, meant to be included in an AVA (additional
valuation adjustment) in an FRTB terminology.

109 / 244

Economic Capital

The economic capital (EC) of the bank is its resource devoted tocope with losses beyond their expected levels that are already takencare of by reserve capital (RC).
Basel II Pillar II defines economic capital as the 99% value-at-risk ofthe negative of the variation over a one-year period of core equity tierI capital (CET1), the regulatory metric that represents the wealth of
the shareholders within the bank.

110 / 244

Recently, the FRTB required a shift from 99% value-at-risk to 97.5%
expected shortfall.
In our setup, capital depletions correspond to the trading loss processL.

111 / 244

Accordingly, also accounting for discounting (and recalling that L0 is
the F reduction of L):

Definition 5
Our reference definition for the (discounted) economic capital of the bankat time t is the (Ft , P) conditional 97.5% expected shortfall off t (L).
(Le0t+1 − Le0t ), which we denote by ES
Solvency II introduces a further modification of economic capital,
which is required to be in excess of the risk margin (RM), i.e. of the
KVA (cf. (13)). This modification is considered later.

112 / 244

For the purpose of economic capital and cost of capital computations,the trading loss process L of the bank can be considered as an
exogenous process ((G, Q) martingale without jump at τ , by Lemma
1).
f t for ES
f t (L), and ESt for ESt (L), theAccordingly we just write ES
f t (L).
undiscounted version of ES

Lemma 3
ES is nonnegative.

113 / 244

KVA as a Risk PremiumCounterparty default losses, as also funding payments, arematerialities for default if not paid, hence the CVA and the FVA are
liabilities (or “contra-assets”) to shareholders.
In contrast, KVA payments are at the discretion of the bankmanagement and released to bank shareholders themselves.
Accordingly:

Assumption 10
The risk margin is loss-absorbing, hence part of economic capital.

114 / 244

Corollary 5
Shareholder capital at risk (SCR) is the difference between the economic
capital (EC) of the bank and its risk margin (RM), i.e.
SCR = EC − RM.

(27)

Assumption 11
An exogenous and constant hurdle rate h prevails, in the sense that bankshareholders are constantly maintained by the KVA payments on an“efficient frontier” such that, at any time t“Shareholder instantaneous average returnt = h × SCRt .”(28)

115 / 244

In practice the level of compensation required by shareholders on theircapital at risk is driven by market considerations. Typically, investors
in banks expect a hurdle rate h of about 10% to 12%.
We assume a constant h for simplicity.
An endogenous and stochastic hurdle rate would arise in a model ofcompetitive equilibrium, where different banks compete for clients.
As opposed to our setup where only one bank is considered.

116 / 244

In view of Lemma 2 and Corollary 5, where RM = KVA holds at all
times by (13), and since Le is a (G, Q) martingale by Lemma 1, theinformal statement (28) is formulated in mathematical terms by therequirement that◦] ) has a (G, Q) drift given as
(−KVA

(29)

f − KVA),]
the time-integrated process h (ECassumed to define a unique KVA process in S2◦This includes that the KVA process is defined until τ̄ and without jumpat τ .

117 / 244

However, the KVA equation (29) is only preliminary if EC there isjust meant as ES, which would then be forgetful of a consistency
condition SCR ≥ 0.
This is fixed in the next section by pushing EC above ES until theconstraint is satisfied.

118 / 244

The KVA Constrained Optimization Problem

Assume that, for any tentative economic capital process C in asuitable Hilbert space L2 of square integrable processes containing S2
and the process ES, the equation (cf. (29))(−Ke ◦ ) has a (G, Q) drift given as the time-integrated process h(Ce − Ke )defines a unique process K = K (C ) in S2◦ .

119 / 244

Definition 6
The set of admissible economic capital processes is defined as
C = {C ∈ L2 ; C ≥ max K (C ), ES },(31)

where (b) C ≥ ES is the risk acceptability condition and (a) C ≥ K (C ) is
the self-consistency condition.

120 / 244

The KVA Constrained Optimization Problem

In view of (30) and (31), the natural guess for the smallest andcheapest admissible economic capital process is thenEC = max(ES, KVA),(32)

for a process KVA in S2◦ such that◦] ) has a (G, Q) drift given as the(−KVA
f KVA)] − KVA] .
time-integrated process h max(ES,(33)→ The discounted KVA is a (G, Q) supermartingale.

121 / 244

Remark 4
In the case of perfect clean and CA hedges where the process L (hence L0 )is constant, then ES vanishes and KVA = 0 obviously solves (33) in S2◦ .

122 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

Given the actual (incremental) derivative porfolio of a bank, theabove can be applied to the version of the portfolio that would be
run-off by the bank from time 0 onward until its final maturity T .
The ensuing XVA numbers are interpreted as the amounts
CA = CVA + FVA to maintain on the reserve capital (RC) accountand KVA to maintain on the risk margin (RM) account, which would
allow the bank to go into run-off in line with shareholder interest.

124 / 244

Such a “soft landing option” is key from a regulator point of view, as
it guarantees that the bank should not be tempted to go into
snowball or Ponzi kind of schemes where always more trades areentered for the sole purpose of funding previously entered ones.
Moreover, since we rely on a dynamic analysis, this possibility, for abank respecting the balance conditions (13) for CVA, FVA, and KVA
as per (23), (24), and (33), of going run-off in line with shareholderinterest, is granted not only from time 0 onward, but from any future
time onward, as long as no new deals occur to the bank.

125 / 244

A new trade has two impacts: it triggers a wealth transfer fromshareholders to bondholders and alters the risk profile of the portfolio.
This is reflected by a jump “∆·” of the balance sheet, from thebalance sheet related to the endowment (pre-trade portfolio) at thetime t where the new deal is considered, to the balance sheet related
to the portfolio including the new deal at time t (both portfolios
being assumed held on a run-off basis).

126 / 244

Hence the balance conditions (13) and the associated soft landing
option of the bank are impaired, unless the missing RC and RM
amounts are sourced from the client of the deal in order to restore
them.
→ Assumption 1 implies∆RC = ∆CA, ∆RM = ∆KVA,
for ∆XVAs computed on an incremental run-off basis based on (22)and (33).

127 / 244

→ The all-inclusive XVA add-on to the entry price for a new deal, calledfund transfer pricing (FTP), isFTP = ∆CA + ∆KVA = ∆CVA + ∆FVA + ∆KVA.

(34)

Obviously, the endowment has a key impact on the FTP of a new
trade. For instance, it can happen that a new deal is risk-reducing withrespect to the pre-existing portfolio, in which case FTP < 0.

128 / 244

The preservation of the balance conditions in between and throughout
deals yields a sustainable strategy for profits retention, which isalready the key principle behind the Eurozone Solvency II insurance
regulation.

129 / 244

From this “soft landing” perspective it is natural to perform the XVA
computations under the following assumption, in line with a run-off
procedure where market risk is first hedged out, but we conservatively
assume no XVA hedge, and the portfolio is then let to amortize until
its final maturity T :Assumption 12
We assume a perfect clean hedge by the clean traders, i.e. L = Lca , and no
CA hedge, i.e. H = 0.

130 / 244

As it then immediately follows from Lemma 1:

Corollary 6
We haveg + Ce◦ + Fe◦ .
Le = Leca = CA

(35)

→ The process L that is used as input to capital and KVA computations
is the output of the CA computations, making the XVA problem as awhole self-contained.
cf. Definition 5 and (33), where ES = ES(L)131 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

“Recursive” (G, Q) XVA equations as per Duffie and Singleton (1999)
or Collin-Dufresne, Goldstein, and Hugonnier (2004)
Solved by reduction to (F, P), in the present invariance timeframework for τ

133 / 244

SpacesAssuming that the pre-intensity γ of τ is F predictable (without loss ofgenerality by reduction), we denote by:
S2 , the space of càdlàg G adapted processes Y over [0, τ̄ ] such that
Z T R
i
h
s
2
e 0 γu du 1{s<τ } d(Ys∗ )2 < ∞,E Y0 +(36)0

where Yt∗ = sups∈[0,t] |Ys |;S2◦ , the subspace of the processes Y in S2 such that Y is without
jump at τ on {τ < T } and YT = 0 on {T < τ };134 / 244

S2• , the subspace of the processes Y in S2 such that Yτ̄ = 0;
S20 , the space of càdlàg F adapted processes Y 0 over [0, T ] such thathi
E0 sup (Yt0 )2 < ∞(37)t∈[0,T ]and YT0 = 0;

135 / 244

L2 , the space of G progressively measurable processes X over [0, T ]such thatZ T R s γu du(38)E1{s<τ } Xs2 ds < +∞;e 0
0L02 , the space of F progressively measurable processes X 0 over [0, T ]such thatZ T 0 2 
0(39)E(Xt ) dt < +∞.
0

136 / 244

KVA in the Case of a Default-Free Bank

Note that the primary reason for the KVA to exist is the default of
the bank clients, as opposed to the default of the bank itself
which on the other hand is the key of the contra-liabilities related
wealth transfer issue.

In this part we suppose the bank default free, i.e.
τ = +∞ , (F, P) = (G, Q) and γ = 0.
This is then extended to the case of a defaultable bank in the next part.

137 / 244

At that stage in this part we use the “·0 ” notation, not in the sense of
reduction (as F = G), but simply in order to distinguish the equationsin this part, where F = G, from the ones in the next part, where
F 6= GThe data of this subsection will then be interpreted a posteriori as the
reductions of the corresponding data in the next subsection.

138 / 244

Given C 0 ≥ ES representing a putative economic capital process for thebank, consider the following BSDEs (cf. (30) and (33) when τ = +∞):
Kt0 = E0t

Z T

hCs0 − (rs + h)Ks0 ds , t ∈ [0, T ],

(40)

t

KVA0t = E0t

Z Th max(ESs , KVA0s ) − (rs + h)KVA0s ds , t ∈ [0,(41)T]t

to be solved for respective processes K 0 and KVA0 .

139 / 244

Lemma 4
Assuming that r is bounded from below and that r , C 0 , and ES are in L02 ,then the BSDEs (40) and (41) are well posed in S20 , where well-posedness
includes existence, uniqueness and comparison. We have, for t ∈ [0, T ],KVA0t = hE0t

Z T

Rs

e − t (ru +h)du max(ESs , KVA0s )ds.

(42)

t

Proof. By application of monotonic coefficient BSDE results (seee.g. Kruse and Popier (2016, Sect. 4)).

140 / 244

Proposition 4
Assuming that r is bounded from below and that r and ES are in L02 , we
have:(i) EC0 = min C 0 , KVA0 = minC 0 ∈C 0 K 0 (C 0 );(ii) The process KVA0 is nonnegative and it is nondecreasing in h.
Proof. By applications of BSDE comparison theorems (see e.g. Kruse andPopier (2016, Proposition 4)).

141 / 244

KVA in the Case of a Defaultable Bank

In the case of a defaultable bank, “·0 ” now denoting reduction, then, by
the results of Crépey and Song (2017b):
For any C ∈ L2 , we have C 0 ∈ L02 and the (G, Q) BSDE (30) in S2◦ isequivalent to the (F, P) BSDE (40) in S20 through the correspondence
K = (K 0 )τ − on [0, τ̄ ];Assuming ES in L02 , the (G, Q) KVA BSDE (33) in S2◦ is equivalent
to the (F, P) KVA0 BSDE (41) in S20 through the correspondence
KVA = (KVA0 )τ − on [0, τ̄ ].

142 / 244

Hence, by application of Lemma 4 and Proposition 4 through the above
correspondences:

Lemma 5
Assuming that r is bounded from below and that r , C 0 , and ES are in L02 ,then the (G, Q) linear BSDEs (30) for K = K (C ) and the (G, Q) KVA
BSDE (33) are well posed in S2 , where well-posedness includes existence,uniqueness and comparison.

143 / 244

Theorem 1
Assuming that r is bounded from below and that r and ES are in L02 :(i) EC = min C, KVA = minC ∈C K (C );(ii) The process KVA is nonnegative and it is nondecreasing in h.

144 / 244

Contra-assets and Contra-liabilities

The counterparty exposure and funding cumulative cash flow streams
Y = C and F (recall Assumption 12 set H = 0) are given as G finite
variation processes.
C ◦ and F ◦ can be assumed to be F finite variation processes, without
loss of generality by reduction.

145 / 244

Regarding the funding cash flows, we assume more specifically:
dFt◦ = ft (FVAt )dt until τ,(43)

for some predictable coefficient (random function) f .
A structure (43) for F is a slight departure from our abstract setup,
where, for simplicity of presentation in a first stage, F was introduced
as an exogenous process.

146 / 244

But, as already found in the one-period setup or in thecontinuous-time example 2, the dependence of F on the FVA is only
semi-linear (i.e. f in (43) is Lipschitz or monotonous) in practice.
Provided the corresponding FVA fixed-point problem is well-posed,
one can readily check, by revisiting all the above, that suchdependence does not affect any of the qualitative conclusions in theabove.

147 / 244

Lemma 6
For C and F thus specialized, the CVA and FVA equations (23) and (24)in S2◦ are equivalent to the following equations in S20 :CVA0t = E0t

Z Tβt−1 βs dCs◦ , t ∈ [0, T ],(44)

βt−1 βs fs (FVA0s )ds , t ∈ [0, T ],

(45)

t

FVA0t = E0t

Z Ttequivalent through the following correspondence:
CVA = (CVA0 )τ − and FVA = (FVA0 )τ − on [0, τ̄ ].

(46)

Proof. By application of the results of Crépey and Song (2017b)

148 / 244

Example 4
In the setup of Example 2, we have, for 0 ≤ t ≤ τ̄ :dFt◦ = λt (Qt − CAt )+ dt
dFt• = (1 − R)(Qt− − CAt− )+ (−dJt ).

(47)

Hence F ◦ is of the form (43), for ft (y ) = λt (Qt − CVAt − y )+ , and◦+dLt = dLcat = dCAt − rt CAt dt + dCt + λt (Qt − CAt ) dt.

(48)

149 / 244

Example 4 (Cont’d)Assume further that the bank portfolio involves a single client withdefault time denoted by τ1 , that Q(τ1 = τ ) = 0, that the liquidationof a defaulted party is instantaneous and that no derivative cash flows
are due at the exact times τ and τ1 .
Let J and J 1 , respectively R and R1 , denote the survival indicator
processes and the recovery rates of the bank and its client.

150 / 244

Example 4 (Cont’d)Then Q is of the form J 1 Q 1 , where Q 1 is the difference between themark-to-market P of the variation margin provided by the CA desk to
the clean desks and the mark-to-market of the variation margin
provided to the CA desk by the client.
Moreover, for 0 ≤ t ≤ τ̄ ,dCt◦ = (1 − R1 )(Qτ11 )+ (−dJt1 )dCt• = 1{τ ≤τ1 } (1 − R)(Qτ1 )− (−dJt ).

(49)

151 / 244

Proposition 5
In the setup of Example 4, assuming that r is bounded from below andthat the processes r , λ, and λ(J 1 Q 1 − CVA0 )+ are in L02 , and that CVA0in (51) is in S20 , then the CVA and FVA equations (23) and (24) are
well-posed in S2◦ and we have, for 0 ≤ t ≤ τ̄ :CVAt = (CVA0 )τt − and FVAt = (FVA0 )τt − , where for 0 ≤ t ≤ T :

CVA0t = E0t 1{t<τ1 <T } βt−1 βτ1 (1 − R1 )(Qτ11 )+ ;Z T
βt−1 βs λs (Js1 Qs1 − CVA0s − FVA0s )+ ds,
FVA0t = E0t

(50

(51

(52

t

dLt = dCAt − rt CAt dt + (1 − R1 )(Qτ11 )+ (−dJt1 ) + λt (Jt1 Qt1 − CAt )+(53
dt

152 / 244

Proposition 5 (Cont’d)
CLt = Et 1{τ ≤τ1 ∧T } βt−1 βτ1 (1 − R)(Qτ1 )−
{z
}
|

(54)

FTDDVAt+ Et βτ /βt 1{t<τ <T } (Jτ1− Qτ1− − CAτ − )+
|
{z
}

(55)

FDAt





+ Et βt−1 βτ 1{t<τ <T } CVA0τ − + Et βt−1 βτ 1{t<τ <T } FVA0τ(56)
−
|
{z
} |
{z
}
CVACL
t

FVACL
t

153 / 244

Proposition 5 (Cont’d)
CRt = Et 1{t<τ1 ≤τ ∧T } βt−1 βτ1 (1 − R1 )(Qτ11 )+
|
{z
}

(57)

FTDCVAt− Et 1{t<τ ≤τ1 ∧T } βτ1 /βt (1 − R)(Qτ1 )− .
|
{z
}


FTDDVAt

154 / 244

Proof. Under the assumptions of the proposition, the (F, P) FVA0 BSDE(45) is a monotonous coefficient BSDE well-posed in S20 , based on the
results of Kruse and Popier (2016, Sect. 4). In view of Lemma 6, thisproves the CVA and FVA related statements.
The dynamics (53) for L are obtained by plugging into (48) the first line in(49).
The CL and CR formulas (56) and (57) readily follow from (25), (47) and(49) for CL and Definition 2 and (49) for CR.

155 / 244

Proposition 5 is easily extended to bilateral trade portfolios withseveral counterparties.
cf. Albanese, Caenazzo, and Crépey (2017) and (64)-(65) below

156 / 244

Proposition 5 is derived in a pure valuation perspective.
In most other former XVA references in the literature, XVA equations
are based on hedging arguments.
Most previous XVA works were not considering KVA yet.
Under our approach, the KVA is the risk premium for the market
incompleteness related to contra-assets.
Hence, for consistency, our KVA treatment requires a pure valuation(as opposed to hedging) view on contra-assets.

157 / 244

Formula (57) is symmetrical, i.e. consistent with the law of one price,in the sense that (FTDCVA − FTDDVA) corresponds to thenegative of the analogous quantity considered from the point of viewof the counterparty.

158 / 244

It only involves the first-to-default CVAs and DVAs, where the default
losses are only considered until the first occurrence of a default of thebank or its counterparty in the deal.
This is consistent with the fact that later cash flows will not be paid inprinciple.

159 / 244

Since the presence of collateral has a direct reducing impact on
FTDCVA/DVA, this formula may give the impression thatcollateralization achieves a reduction in counterparty risk at no costto either the bank or the clients.

160 / 244

However, in the present incomplete market setup, the value CR fromthe point of view of the bank as a whole ignores the misalignement ofinterest between the shareholders and the creditors of a bank.

161 / 244

Proposition 5 analyses the cost of counterparty risk to shareholders(CA) and the wealth transfer (CL) triggered from the shareholders to
the creditors by the impossibility for the bank to hedge its own
jump-to-default exposure.

162 / 244

Due to the latter and to the impossibility for the bank to replicate
counterparty default losses, these contra-liabilities (CL) as well as thecost of capital (KVA) are material to shareholders and need to bereflected in entry prices on top of the fair valuation (CR) of
counterparty risk.

163 / 244

Only the fluctuations of L matter in economic capital calculations,hence the (unknown) value of L0 is immaterial in all XVA
computations.

164 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

Even though our setup includes the default of the bank itself, which isthe essence of the contra-liabilities related wealth transfer issue, we
end up with unilateral CVA, FVA and KVA formulas such as (51),(52) and (41) pricing the related cash flows until the end of times T(as opposed to τ̄ = τ ∧ T ).
And these equations only involve the original discount factor β, without
any credit spread.

166 / 244

This is indeed what follows from a careful analysis of the wealthtransfers involved.
However this also makes the ensuing XVAs more expensive than thebilateral XVAs that appear in most of the related literature.

167 / 244A unilateral CVA is actually required for being in line with theregulatory requirement that reserve capital should not diminish as aneffect of the sole deterioration of the bank credit spread.

168 / 244

But a bilateral FVA already satisfies the regulatory monotonicity
requirement
Essentially, as, when the bank credit spread deteriorates, the shortest
duration of a bilateral FVA is compensated by the higher funding
spread.

169 / 244

And the KVA is not concerned by this requirement.
Actually, a unilateral KVA might arguably be unjustified, with regard to
the fact that bank insolvency means depletion of the whole economic
capital of the bank, which includes the risk margin. Hence the notionof transfer of the residual risk margin to creditors at bank default
would be pointless.
However, the default of a bank does not mean insolvency, but illiquidity
mainly.

170 / 244

From Unilateral to Bilateral KVA

Assuming all the risk margin already gone at time τ̄ through an additionalmodel feature, such as an operational loss that would occur at τ andtrigger instantaneous depletion of economic capital, would result in thefollowing modified KVA equation in S2• :◦] ) has a (G, Q) drift given as the(−KVA
f KVA)] − KVA] ,time-integrated process h max(ES,(58)

171 / 244

i.e., in the continuous time setup,Z τ̄
Rs
KVAt = hEt
e − t (ru +h)du max(ESs , KVAs )ds, t ∈ [0, τ̄ ],

(59)

t

or, in an an equivalent (F, P) formulation, KVA = (KVA0 )τ on [0, τ̄ ],where (compare with (42), noting in particular the “+γu ” in the discountfactor in (60))
KVA0t = hE0t

Z T

Rs

e − t (ru +h+γu )du max(ESs , KVA0s )ds, t ∈ [0, T ].

(60)

t

172 / 244

From Unilateral to Bilateral FVA

A bilateral FVA, which already satisfies the regulatory monotonicity
requirement on the related reserve capital, might be advocated asfollows.
Assume for the sake of the argument that the portfolio of thedefaulted bank with clients is unwounded with risk-freecounterparties, called novators.

173 / 244

The residual amount of CVA reserve capital is required by thenovators to deal with the residual counteparty risk on the deals.
But the residual amount of FVA reserve capital is useless to thenovators.

174 / 244

In view of this one could decide that, upon bank default, the residual
FVA capital reserve flows back into equity capital and not to creditors.
For formalizing this mathematically, one needs disentangle the CA
desk into a CVA desk and an FVA desk, each endowed with their own
reserve capital account (and hedge).

175 / 244

This would result in an FVA equation stated in S2• as
] t = Et Feτ̄◦ − Fet◦ , t ≤ τ̄ ,
FVA

(61)

instead of the FVA equation (24) in S2◦ .

176 / 244

That is, in the continuous-time setup: (compare with (45))
Z τ̄
FVAt = Et

βt−1 βs fs (FVAs )ds, t ∈ [0, τ̄ ],

(62)

t

or, equivalently, FVA = (FVA0 )τ on [0, τ̄ ], where
FVA0t = E0t

Z T

Rs

e − t (ru +γu )du fs (FVA0s )ds, t ∈ [0, T ].

(63)

t

Note again the blended discount factor in (63), as opposed to therisk-free discount factor β in (45).

177 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

Next we account for the additional FVA reduction provided by thepossibility for a bank to post economic capital, on top of reserve
capital already included in the above, as variation margin.
Note that, in principle, uninvested capital (UC) could be used for VM
as well, but since UC is not known and could as well be zero in the
future, capital is conservatively taken here as (RC+EC).

179 / 244

Accounting for the use of EC as VM, the VM funding needs are
reduced from (Q − CA)+ to (Q − EC(L) − CA)+ .
As a consequence, instead of an exogenous CA value process feeding
the dynamics for L (cf. e.g. (53)), one obtains a FBSDE system
made of a forward SDE for L coupled with a backward SDE for the
CA value process.

180 / 244

iSo, assuming
P i i n counterparties with survival indicator processes J , hence
Q = J Q , using here unilateral CVA vs. bilateral FVA and KVA as apractical trade-off:

L0 = z and, for t ∈ (0, τ̄ ],X
dLt = dCAt +(1 − Ri )(Qτi i )+ (−dJti )

(64)

i



+ λt

X

Jti Qti − ECt (L) − CAt+− rt CAt dt,i

181 / 244whereX

CAt = Et

βt−1 βτi (1 − Ri )(Qτi i )+t<τi <T

{z

|

}

CVAt

Z τ̄
+ Et
t

|

βt−1 βs λs

X

Jsi Qsi − ECs (L) − CAs

+

(65)
ds , 0 ≤ t ≤ τ̄ .

i

{z

FVAt

}

182 / 244

Unless λ = 0, nonstandard coupling between L and CA through theterm ECt (L), which entails the conditional law of the one-year-ahead
increments of L.
Crépey, Élie, Sabbagh, and Song (2017) show that:
This FBSDE for L and CA can be decoupled into an anticipated BSDE(“McKean” ABSDE) for the underlying FVA process;Our previous results are still valid provided one replaces (Q − CA)+ by(Q − EC(L) − CA)+ everywhere.

183 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

The above XVA approach can be implemented by means of nested
Monte Carlo simulations for approximating the loss process L required
as input data in the KVA computations. Contra-assets (andcontra-liabilities if wished) are computed at the same time.
Since one of our goals in the numerics is to emphasize the impact on
the FVA of the funding sources provided by reserve capital andeconomic capital, we consider the FBSDE (64)–(65) which accounts
for the use of EC (on top of RC) as VM.

185 / 244

Let
(0)

FVAt

Z τ̄
= Et
t

βt−1 βs λs (X

Jsi Qsi )+ ds,iwhich corresponds to the FVA accounting only for the
re-hypothecation of the variation margin received on hedges, but
ignores the FVA deductions reflecting the possible use of reserve andeconomical capital as VM.

186 / 244

Picard iteration

L(0) = z, FVA(0) as above, CA(0) = CVA + FVA(0) and, for k ≥ 1,(k)

L0 = z and, for t ∈ (0, τ̄ ],
(k)

(k−1)

dLt

= dCAt

+ λt

X

(k−1)

− rt CAt

dt +X(1 − Ri )(Qτi i )+ (−dJti )

i

(66)


(k−1) (k−1) +Jti Qti − max ESt (L(k−1) ), KVAt
− CAt
dt,

i

187 / 244

(k)

CAt

(k)

t
(k)

KVAt

(k)

= CVAt + FVAt where FVAt =Z τ̄X(k−1) (k−1) +Et
βt−1 βs λs
Jsi Qsi − max ESs (L(k) ), KVAs
− CAs
i

Z τ̄
= hEt

Rs

(k−1) e − t (ru +h)du max ESs (L(k) ), KVAs

ds.

t

(67)

188 / 244

Numerically, one iterates (66)–(67) as many times as is required toreach a fixed point within a preset accuracy.
In the case studies we considered, one iteration (k = 1) was foundsufficient.

189 / 244

A second iteration did not bring significant change as
In (64)-(65) the FVA feeds into economic capital only through FVA
volatility and the economic capital feeds into FVA through a capitalterm which is typically not FVA dominated
In (59), in most cases we have that EC = ES. The inequality only
stops holding when the hurdle rate h is very high and the term
structure of EC starts very low and has a sharp peak in a few years,
which is quite unusual for a portfolio held on a run-off basis, asconsidered in XVA computations, which tends to amortize in time.

190 / 244

Economic capital projections

However, going even once through (66)–(67) necessitates theconditional risk measure simulation of ECt (L).
On realistically large portfolios, some approximation is required for thesake of tractability.

191 / 244

The simulated paths of L(1) are used for inferring a deterministic term
structure
ES(1) (t) ≈ ESt (L(1) )(68)

of economic capital, obtained by projecting in time instead ofconditioning with respect to Gt in ES.

192 / 244

Discussion of the economic capital term structure approximation
If a corporate holds a bank payable, it typically has a desire to close
it, receive cash, and restructure the hedge with a par contract (thebank would agree to close the deal as a market maker, charging fees
for the new trade).
Because of this natural selection, a bank is mostly in the receivables
in its derivative business with corporates.

193 / 244

Discussion (Cont’d)Hence, the tail-fluctuations of its loss process L are mostly driven by
the counterparty default events rather than by the volatility of theunderlying market exposure.
Thus, working with a deterministic term structure approximation
ES(1) (t) of economic capital is acceptable.

194 / 244

Discussion (Cont’d)If, by exception, the derivative portfolio of a bank is mostly in thepayables, then all XVA numbers are small and matter much lessanyway195 / 244Symmetric FVA

A similar argument is sometimes used to defend a symmetric FVA (or
SFVA) approach, such as, instead of FVA in (65):Z τ̄
SFVAt = Ettes
βt−1 βs λXJsi Psi ds , 0 ≤ t ≤ τ̄ ,

(69)

i

et
for some VM blended funding spread λcf. Piterbarg (2010), Burgard and Kjaer (2013b), and the discussion inAndersen, Duffie, and Song (2017).

From the FCA/FBA accounting and funds transfer pricing industry
standard to an FVA/FDA accounting and pricing framework.

196 / 244

This corresponds to an FCA/FBA pricing policy, as opposed to our
FVA/FDA approach.
The explicit, linear SFVA formula can be implemented by standard(non-nested) Monte Carlo simulations.

197 / 244

et , the equation yields
For a suitably chosen blended spread λreasonable results in the case of a typical bank portfolio dominated by
unsecured receivables.
However, in the case of a portfolio dominated by unsecured payables,
this equation could yield a negative FVA, i.e. an FVA benefit,
proportional to the own credit spread of the bank, which is not
acceptable from a regulatory point of view.

198 / 244

Asymmetric FVA is more rigorous and has been considered in
Albanese and Andersen (2014), Albanese, Andersen, and Iabichino(2015), Crépey (2015), Crépey and Song (2016), Brigo andPallavicini (2014), Bielecki and Rutkowski (2015), and Bichuch,
Capponi, and Sturm (2017).
In this work, we improve upon such asymmetric FVA models byaccounting for the funding source provided by economic capital(cf. (65)).

199 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

We present two XVA case studies on fixed-income andforeign-exchange portfolios. Toward this end we use the market andcredit portfolio models of Albanese, Bellaj, Gimonet, and Pietronero(2011) calibrated to the relevant market data.
We use nested simulation with primary scenarios and secondary
scenarios generated under the risk neutral measure calibrated toderivative data using broker datasets for derivative market data.

201 / 244

All the computations are run using a 4-socket server for Monte Carlosimulations, Nvidia GPUs for algebraic calculations and Global
Valuation Esther as simulation software. Using this super-computerand GPU technology the whole calculation takes a few minutes forbuilding the models, followed by a nested simulation time in the orderof about an hour for processing a billion scenarios on a real-life bankportfolio.

202 / 244

Toy Portfolio

We first consider a portfolio of ten USD currency fixed-income swaps on
the date of 11 January 2016 (without initial margins, i.e. for IM = 0).

203 / 244

Toy portfolio of swaps (the nominal ofeach swap is $104 )
Mat.
10y
10y
5y
5y
30y
30y
2y
2y
15y
15y

Receiver Rate
Par 6M
LIBOR 3M
Par 6M
LIBOR 3M
Par 6M
LIBOR 3M
Par 6M
LIBOR 3M
Par 6M
LIBOR 3M

Payer Rate
LIBOR 3M
Par 6M
LIBOR 3M
Par 6M
LIBOR 3M
Par 6M
LIBOR 3M
Par 6M
LIBOR 3M
Par 6M

Credit curves of the bank andits four conterparties

i
3
2
2
3
2
1
1
4
1
4

204 / 244

Introducing financial contracts one after the other in one or the reverse
order in a portfolio at time 0 results in the same aggregated incremental
FTP amounts for the bank, equal to the “time 0 portfolio FTP”, but indifferent FTPs for each given contract and counterparty.

205 / 244

Toy portfolio. Left: XVA values and standard relative errors (SE).
Right: Respective impacts when Swaps 5 and 9 are added last in theportfolio.
UCVA0
(0)
FVA0
FVA0
KVA0
FTDCVA0FTDDVA0$Value471.2373.87
3.87
668.83372.22335.94

SE
0.46%
1.06%
4.3%
N/A
0.46%
0.51%

∆UCVA0
(0)
∆FVA0
∆FVA0
∆KVA0
∆FTDCVA0∆FTDDVA0

Swap 5155.46-85.28-80.13127.5498.49
122.91

Swap 9-27.17-8.81-5.80-52.85-23.83-80.13206 / 244

Representative Portfolio

We now consider a representative portfolio with about 2,000
counterparties, 100,000 fixed income trades including swaps, swaptions,
FX options, inflation swaps and CDS trades (IM = 0).
XVA
UCVA0
(0)
FVA0
FVA0
KVA0
FTDCVAFTDDVA$Value
242 M
126 M
62 M
275 M
194 M
166 M

207 / 244

Left: FVA blended funding curve computed from the ground up basedon capital projections.
Right: Term structure of economic capital compared with the term
structure of KVA.

Return to FVA

Return to KVA

Return to SCR

208 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

CVA and FVA: Comparison with the BK Approach

Burgard and Kjaer (2011b, 2013a, 2017) repeatedly (and rightfully) saythat only pre-default cash-flows matter to shareholders. For instance,
quoting the first paragraph in the second reference:“Some authors have considered cases where the post-default cash
flows on the funding leg are disregarded but not the ones on thederivative. But it is not clear why some post default cashflows
should be disregarded but not others”,

towhich we subscribe fully.

210 / 244

The introduction of their now classical “(funding) strategy I :semi-replication with no shortfall at own default” (see e.g. (Burgard
and Kjaer 2013a, Section 3.2)) seems to be in line with the idea,
which we also agree with (see Assumption 7 and the comment
following it), that a shortfall of the bank at its own default does notmake much sense and should be excluded from a model.

211 / 244

However:Being rigorous with the first principle above implies that the valuation
jump of the portfolio at the own default of the bank should bedisregarded in the shareholder cash flow stream. But their
computations do not exclude this cash flow;
We would not say that they really exclude shortfall at bank own
default.

212 / 244

Reviewing the funding strategies in Burgard and Kjaer (2017, Section4), considering for simplicity the special case with sB = 0 there, i.e. apure CVA setup:
Their strategy III, claimed to imply a unilateral CVA as per Albaneseand Andersen (2014) (i.e. (50)–(51) as made more precise here),
does in fact not.
Duly accounting for the transfer of the residual reserve capital fromshareholders to creditors at the bank default time τ (cf. (22)), thefunding strategy that does so is simply funding and investing at r = 0(having assumed sB = 0).

213 / 244

Reviewing the funding strategies in Burgard and Kjaer (2017, Section4) (Cont’d):Their respective strategies I and II not only do not imply the claimed
XVA formulas.
Besides, they do not satisfy the second part in Assumption 7.
Unless in their notation V ≥ 0, respectively Vb ≥ 0, i.e. in our notation
MtM ≥ 0, respectively MtM − CVA ≥ 0.

214 / 244

KVA: Comparison with the GK Approach

In Green et al. (2014) and as also discussed in some theoreticalactuarial literature (see Salzmann and Wüthrich (2010, Section 4.4)),
the KVA is treated as a liability.

215 / 244

Viewing the KVA as a liability, hence non loss-absorbing, results in
EC = SCR = ES (as opposed to (27) in our setup), and therefore
f instead of h(ECf − KVA)] in the KVA equation (29).
hESThis implies r instead of (r + h) as discount rate in the KVA formula(42) (where KVA0 and KVA coincide before τ ) or its bilateral analog(59).

216 / 244

Moreover, if the KVA is viewed as a liability, forward starting
one-year-ahead fluctuations of the KVA must be simulated for
economic capital calculation. This makes it intractable numerically,
unless one switches from economic capital to regulatory capital in the
KVA equation.
Using regulatory instead of economic capital is then motivated bypractical considerations but is less self-consistent. It loses theconnection, established from shareholder-optimization principles in
the above, whereby the correct KVA input is the CA desk loss process
L = Lca as per (35).

217 / 244

It also leads to unachievement of the corresponding XVA theory:
Viewing the KVA as a liability means that KVA payments contributeto the trading profit-and-loss of the bank. But dividends are meant toremunerate risk, i.e. unhedged losses. Hence the XVA loop does not
close.

218 / 244

In addition, Green et al. (2014) derive their KVA equation in a
replication framework, whereas the main motivation for capitalrequirements, such as CVA reserve capital to be held against
counterparty default losses, is that credit markets are incomplete andhedging is not possible.
A KVA equation similar to the one in Green et al. (2014) is derived in
an expectation setup in Elouerkhaoui (2016).

219 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

Counterparty Risk,with its funding and capital implications, is at the origin of all XVAs:
CVA Credit valuation adjustment
The value you lose due to the defaultability of yourcounterparties
DVA Debit valuation adjustment
The value your counterparties lose due to your own
defaultability
The symmetric companion of the CVA
The value you gain due to your own defaultability?
(2011 DVA debate)

221 / 244

FVA Funding valuation adjustment
Cost of funding variation and initial margin: MVA
merged with FVA in these slides to spare one “VA”
But what about the Modigliani-Miller theorem??
(2013 FVA debate )DVA2 Funding windfall benefit at own default
KVA Cost of capital
The price for the bank of having to reserve capital atrisk (ongoing KVA debate )222 / 244

CVA−DVA+FVA−DVA2 [+KVA]: The XVA debates

FVA and DVA2 cash flows NPV-match each other→

CVA-DVA yields the fair, symmetrical adjustment between two
counterparties of equal bargaining power
But “Contra-liabilities” DVA and DVA2 are only a benefit to thecreditors of the bank, whereas only the interest of shareholders
matters in bank managerial decisions→ DVA and DVA2 should be ignored in entry prices→

CVA+FVA

223 / 244

KVA and Funds Transfer Price

Moreover, counterparty default losses cannot be replicated and a bankmust reserve shareholder capital to cope with residual risk
Shareholders that put capital at risk deserve a remuneration at a
hurdle rate, which corresponds to the KVA→

FTP=CVA+FVA+KVA

224 / 244

Connection with the Modigliani and Miller (1958) Theorem

The Modigliani-Miller theorem includes two key assumptions.
One is that, as a consequence of trading, total wealth is conserved.
The second assumption is that markets are complete.

In our setup we keep the wealth conservation hypothesis but we lift
the completeness.
Hence the conclusion of the theorem, according to which the fairvaluation of counterparty risk to the bank as a whole should notdepend on its funding policy, is preserved.

225 / 244

However, due to the incompleteness of counterparty risk, derivatives
trigger wealths transfers from bank shareholders to creditors
The interests of bank shareholders and creditors are not aligned witheach other

Which, in the case of a market maker such as a bank, can only becompensated by add-on to entry prices

226 / 244

More precisely, quoting Villamil (2008):
In fact what is currently understood as the Modigliani-Miller
Proposition comprises four distinct results from a series of
papers (1958, 1961, 1963). The first proposition establishes
that under certain conditions, a firms debt-equity ratio does
not affect its market value. The second proposition establishes
that a firms leverage has no effect on its weighted average
cost of capital (i.e., the cost of equity capital is a linear
function of the debt-equity ratio). The third proposition establishes that firm market value is independent of its dividend policy.
The fourth proposition establishes that equity-holders are indifferent
about the firms financial policy.

227 / 244

The proof of the fourth proposition is based on the ability ofshareholders to redeem all debt of the bank in order to prevent wealthtransfers to creditors.

228 / 244

However:Redeeming the debt means hedging its own default, which is notpossible for a bank.
Banks are special firms in that they are intrinsically leveraged andcannot be transformed into a pure equity entity.
This is also related to an argument of scale.
Banks liabilities are overwhelming with respect to all other wealthnumbers.
It has been estimated that if all European banks were to be required tohave capital equal to a third of liabilities, the total capitalization ofbanks would be greater than the total capitalization of the entire equity
market as we know it today.

229 / 244

Hence:Shareholders cannot redeem all debt of the bank.
The assumption of the fourth proposition of the Modigliani-Miller
theorem does not apply to a bank.

230 / 244

Quoting the conclusion of Modigliani and Miller (1958)“Theseandotherdrastic
simplifications
have
been
necessary
in
order
to
come
to
grips
Having served theirwith
theproblematall.

purpose they can now be relaxed in the direction of greater realism
and relevance, a task in which we hope others interested in thisarea will wish to share.”231 / 244

And Miller (1988) in The Modigliani-Miller Proposition after Thirty
Years“Showing what doesn’t matter can also show, by implication,
what does. ”

232 / 244

Outline
1

Introduction

2

Preliminary Approach in a Static Setup

3

Continuous-Time Setup

4

Derivation of the CVA and FVA Equations

5

Derivation of the KVA Equation

6

Incremental XVA Approach

7

XVA Equations Well-Posedness and Comparison Results

8

Unilateral Versus Bilateral XVAs

9

Using Economic Capital as Variation Margin

10

The XVA Algorithm

11

Case Studies

12

Comparison with the BK and GK Approaches

13

Epilogue

14

Perspectives

XVAs represent a switch of paradigm in derivative
management

From hedging to balance-sheet optimization
Derivative portfolio optimization for a market makerFixes prices, not quantities!

234 / 244

Theoretical Issues

Derivative portfolio optimization for a market maker
Endogenizing the hurdle rate h by introducing competition between
banks
CCP auction price discovery process
XVA model risk / uncertainty quantification issue“Prudent valuation” through “Additional valuation adjustment” (AVA)

235 / 244

Algorithmic and computational challenges

Heavy computations at the portfolio level
Yet needs accuracy so that incremental XVA computations are not in
the numerical noise of the machinery
Using GPU programming and nested Monte Carlo for portfolio-wide
conditional risk measure computations
Using machine learning techniques for solving the resulting
high-dimensional non-convex XVA optimization problems
Multi-level MC, parareal methods, quasi-regression schemes,..

236 / 244

Albanese, C. and L. Andersen (2014).
Accounting for OTC derivatives: Funding adjustments and there-hypothecation option.
Working paper available athttps://papers.ssrn.com/sol3/papers.cfm?abstract id=2482955.
Albanese, C., L. Andersen, and S. Iabichino (2015).
FVA: Accounting and risk management.
Risk Magazine, February 64–68.
Albanese, C., T. Bellaj, G. Gimonet, and G. Pietronero (2011).
Coherent global market simulations and securitization measures forcounterparty credit risk.
Quantitative Finance 11 (1), 1–20.
Albanese, C., S. Caenazzo, and S. Crépey (2017).
Credit, funding, margin, and capital valuation adjustments for bilateral
portfolios.
Probability, Uncertainty and Quantitative Risk 2 (7), 26 pages.
Available at http://rdcu.be/tHKo.
237 / 244

Andersen, L., D. Duffie, and Y. Song (2017).
Funding value adjustments.
First revised version available at https://ssrn.com/abstract=2746010.
Bichuch, M., A. Capponi, and S. Sturm (2017).
Arbitrage-free XVA.
Mathematical Finance.
First published online on 18 April 2017 (preprint version available atssrn.2820257).
Bielecki, T. R. and M. Rutkowski (2015).
Valuation and hedging of contracts with funding costs andcollateralization.
SIAM Journal on Financial Mathematics 6, 594–655.
Brigo, D. and A. Pallavicini (2014).
Nonlinear consistent valuation of CCP cleared or CSA bilateral tradeswith initial margins under credit, funding and wrong-way risks.
Journal of Financial Engineering 1, 1–60.
Burgard, C. and M. Kjaer (2011a).
238 / 244

In the Balance.
Risk, 72–75.
Burgard, C. and M. Kjaer (2011b).
In the balance.
Risk Magazine, October 72–75.
Burgard, C. and M. Kjaer (2011c).
Partial differential equation representations of derivatives withcounterparty risk and funding costs.
The Journal of Credit Risk 7, 1–19.
Burgard, C. and M. Kjaer (2013a).
Funding costs, funding strategies.
Risk Magazine, December 82–87.
Preprint version available at https://ssrn.com/abstract=2027195.
Burgard, C. and M. Kjaer (2013b).
Funding Strategies, Funding Costs.
Risk Magazine, December 82–87.
239 / 244

Burgard, C. and M. Kjaer (2017).
Derivatives funding, netting and accounting.
Risk Magazine, March 100–104.
Preprint version available at https://ssrn.com/abstract=2534011.
Castagna, A. and F. Fede (2013).
Measuring and Managing Liquidity Risk.
Wiley.
Collin-Dufresne, P., R. Goldstein, and J. Hugonnier (2004).
A general formula for valuing defaultable securities.
Econometrica 72 (5), 1377–1407.
Crépey, S. (2015).
Bilateral counterparty risk under funding constraints. Part I: Pricing,followed by Part II: CVA.
Mathematical Finance 25 (1), 1–22 and 23–50.
First published online on 12 December 2012.
Crépey, S., R. Élie, W. Sabbagh, and S. Song (2017).
240 / 244

When capital is a funding source: The XVA Anticipated BSDEs.
Working paper available at https://math.maths.univ-evry.fr/crepey.
Crépey, S. and S. Song (2016).
Counterparty risk and funding: Immersion and beyond.
Finance and Stochastics 20 (4), 901–930.
Crépey, S. and S. Song (2017a).
Invariance times.
The Annals of Probability .
Forthcoming.
Crépey, S. and S. Song (2017b).
Invariance times transfer properties and applications.
Working paper available at https://math.maths.univ-evry.fr/crepey.
Duffie, D. (2010).
How big banks fail and what to do about it.
Princeton University Press.
Duffie, D. and M. Huang (1996).
241 / 244

Swap rates and credit quality.
Journal of Finance 51, 921–950.
Duffie, D. and K. Singleton (1999).
Modeling term structures of defaultable bonds.
Review of Financial Studies 12, 687–720.
Elouerkhaoui, Y. (2016).
From FVA to KVA: including cost of capital in derivatives pricing.
Risk Magazine, March 70–75.
Green, A., C. Kenyon, and C. Dennis (2014).
KVA: capital valuation adjustment by replication.
Risk Magazine, December 82–87.
Preprint version “KVA: capital valuation adjustment” available atssrn.2400324.
Hull, J. and A. White (2012).
The FVA debate, followed by The FVA debate continued.
Risk Magazine, July 83–85 and October 52.
Kruse, T. and A. Popier (2016).
242 / 244

BSDEs with monotone generator driven by Brownian and Poisson noises
in a general filtration.
Stochastics: An International Journal of Probability and Stochastic
Processes 88 (4), 491–539.
Modigliani, F. and M. Miller (1958).
The cost of capital, corporation finance and the theory of investment.
Economic Review 48, 261–297.
Piterbarg, V. (2010).
Funding beyond discounting: collateral agreements and derivatives
pricing.
Risk Magazine 2, 97–102.
Salzmann, R. and M. Wüthrich (2010).
Cost-of-capital margin for a general insurance liability runoff.
ASTIN Bulletin 40 (2), 415–451.
Villamil, A. (2008).
The Modigliani-Miller theorem.
In The New Palgrave Dictionary of Economics.
243 / 244

Available at http://www.econ.uiuc.edu/ avillami/coursefiles/PalgraveRev ModiglianiMiller Villamil.pdf.

244 / 244